# Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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### Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed

This is one of the exercises on Wald's General Relativity: Chapter 8, Problem 8.b Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed. (Hint: ...
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### Is a Klein-bottle-like topology allowed for GR?

As the spacetime of the universe seems to be quite flat, a torus topology comes mind easily. How about others? Is it issue if manifold is non-orientable? I see challenges to find 3- or 4-Klein-bottle-...
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### All Chern numbers as mapping degree?

The Chern number of Haldane model can be interpreted as a mapping degree from $T^2$ (1BZ) to $S^2$ (Bloch sphere). The question is whether all the Chern numbers can be interpreted in this way. In ...
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### Allowed Topologies for General Relativity

Studying the ADM formulation of General Relativity the ADM splitting comes out from the assumption that the spacetime is globally hyperbolic. From that assumption thanks to Geroch's theorem, it is ...
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### Topological proof of spin-statistics theorem confusion

I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as: This is ...
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### Does the total Zak phase always sum to zero?

In 2D, the sum of the Chern numbers over all bands is zero. However, this result relies on the ability to define a Berry curvature, which is only possible in $d \geq 2$ dimensions. In 1D it is ...
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### What is the topology of sine-Gordon equation?

In one pdf on solitons, I am finding the following written For the sine-Gordon theory, it is much better to think of $\phi$ as a field modulo $2\pi$, i.e. as a function $\phi: R \rightarrow S_{1}$. ...
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